Basic knowledge on vectors and Euclidean space of dimension 2 and 3. Basic knowledge on matrices and determinants. Basic knowledge on systems of linear equations. Basic trigonometry.

1. Learning objectives: (PO: a, CGB1) - To solve systems of linear equations and to interpret the results. - To understand the concept of algebraic structure. - To know and understand the notion of vector spaces and their applications. - To understand the notion of bases and coordinates in a vector space. - To understand linear transformations and to represent them by matrices. - To compute the fundamental vector spaces associated to a matrix. - To understand the concept of eigenvalues and eigenvectors of a matrix, and to know their computation and applications. - To compute the QR decomposition of a matrix. - To find an approximate solution to an inconsistent system of linear equation by least-square fitting. - To obtain the singular value decomposition of a matrix. 2. Specific skills: (PO: a, CGB1) - To raise the abstraction level. - To be able to solve real problems using typical linear algebra tools. 3. General skills: (PO: a, CGB1) - To improve the oral and written communication ability using the language and signs of mathematics properly. - To be able to model a real situation described with words using mathematical concepts. - To improve the ability to interpret a mathematical solution and define its limitations and reliability. - To be able to use mathematical software.

1. Matrices - Review of definitions and concepts related to matrices. - Matrix operations. - Transpose. - Inverse. - Determinant. - Sets induced by a matrix. 2. Systems of linear equations - Geometric interpretation of linear systems in R^n. - Existence and uniqueness of solutions. - Matrix methods to solve systems of linear equations. 3. Vector spaces - Vector spaces. - Vector subspaces. - Operations between subspaces. 4. Basis and dimension - Spanning sets. - Basis. Dimension. - Coordinates. 5. Linear transformations - Definition and properties. - Operations between linear transformations. 6. Linear transformation and matrices - Representation of linear transformations using matrices. 7. Change of basis - Change of basis. - Normal form of a linear transformation. 8. Eigenvalues and eigenvectors - Definitions. - Characteristic polynomial and characteristic equation. - Diagonalization. 9. Inner product. Orthogonality - Inner product. - Length and angle. - Orthogonal projection. - Orthogonal complement. 10. Orthogonal basis - Orthogonal sets and orthogonal bases. - Gram-Schmidt process. - QR factorization. 11. The spectral theorem - Diagonalization of symmetric matrices. - Spectral decomposition. 12. Geometry of linear transformations - Reflections. - Contractions and dilations. - Rotations. - Projections. 13. Least squares - The least squares problem. - Geometric interpretation. - Approximation of functions. 14. Pseudoinverse. Singular value decomposition - Pseudoinverse. - Singular value decomposition. - Applications.

Synchronous lecture sessions (3 credits) (PO: a, CGB1) During these sessions we will cover the course topics with the aim of using theory to solve problems. Practicals, working individually and in groups (3 credits) (PO: a, CGB1) During these sessions we will solve exercises of different levels of difficulty.